ferent from a conic
section--it would be a definite class of spiral (called Cotes's spiral).
Again, on an iron filing the force of a single pole might vary more
nearly as the inverse fifth power; and so on. Even when the thing
concerned is radiant in straight lines, like light, the law of inverse
squares is not universally true. Its truth assumes, first, that the
source is a point or sphere; next, that there is no reflection or
refraction of any kind; and lastly, that the medium is perfectly
transparent. The law of inverse squares by no means holds from a prairie
fire for instance, or from a lighthouse, or from a street lamp in a fog.

Mutual perturbations, especially the pull of Jupiter, prevent the path
of a planet from being really and truly an ellipse, or indeed from being
any simple re-entrant curve. Moreover, when a planet possesses a
satellite, it is not the centre of the planet which ever attempts to
describe the Keplerian ellipse, but it is the common centre of gravity
of the two bodies. Thus, in the case of the earth and moon, the point
which really does describe a close attempt at an ellipse is a point
displaced about 3000 miles from the centre of the earth towards the
moon, and is therefore only 1000 miles beneath the surface.

No. 3. Kepler's third law proves that all the planets are acted on by
the same kind of force; of an intensity depending on the mass of the
sun.

The third law of Kepler, although it requires geometry to state and
establish it for elliptic motion (for which it holds just as well as it
does for circular motion), is very easy to establish for circular
motion, by any one who knows about centrifugal force. If _m_ is the mass
of a planet, _v_ its velocity, _r_ the radius of its orbit, and _T_ the
time of describing it; 2[pi]_r_ = _vT_, and the centripetal force
needed to hold it in its orbit is

mv^2 4[pi]^2_mr_
-------- or -----------
_r_ T^2

Now the force of gravitative attraction between the planet and the sun
is

_VmS_
-----,
r^2

where _v_ is a fixed quantity called the gravitation-constant, to be
determined if possible by experiment once for all. Now, expressing the
fact that the force of gravitation _is_ the force holding the planet in,
we write,

4[pi]^2_mr_ _VmS_
----------- = ---------,
T^2 r^2

whence, by the simplest algebra,

r^3 _VS_
------ = ---------.
T^2 4[pi]^2

The